TRANSACTIONS OF THE SECTIONS. 25 



of the resultant of tlie forces -which act normally uponF, 0' the point in which 00 

 is met by the (unknown) neutral axis, A and B the points in which 00 meets the 

 contour of the central nucleus, A' and B' the points in which 00 meets respectively 

 tlie antipolars of A and B *, then this last theorem can be enunciated thus : 



The point C is conjugate to in the involution A.A.', BB'. 



Oonsequently, if is given, we have C linearly, and we construct the neutral 

 axis by drawing through C a sti'aight line parallel to the conjugate direction of 00. 



On a new Construction for the Central Nucleus of a Phne Section. 

 By Professor GirsEPPE Jung (Milan). 



I have the honour to communicate to Section A a new and very easy method of 

 representing the radii of gyration of a given plane (figure F), which appears to be 

 more simple than the known methods of Poinsot, Reye, and Mohr. 



From this representation I deduce a new construction for the central nucleus of 

 F, independent of that of the central ellipse of the figure. Tliis I regard as inter- 

 esting, because of the importance of the central nucleus in the study of the stability 

 of constructions, on account of its remarkable properties with regard to the moment 

 of resistance of the section i&c. (See Oulmann, ' Die graphischo Statik," and the 

 memoirs of which a ?VAK?He has just been given.) 



1. Let be the centre of gravity of F ; AA and BB its principle axes of inertia, 

 (■. c. the axes of the central ellipse E of F ; f and /' (upon AA) the two foci of E ; 

 the circle which has for diameter the major axis A A. Then the radiux of f/yra- 

 tion of F {in the normal direction), ivith respect to any harycentric axis x, is the seff- 

 mentM^l' of the perpendicular dratvn to x from one of the points ff included 

 between the axis x and the circle 0. In fact the circle is the locus of the feet of 

 the perpendiculars let fall from the foci//' upon the tangents to the ellipse E. 



Thus the circle represents the radii of gyration of F (in the normal direction) 

 with respect to all the barycentric axes. If from M' we draw the straight line m 

 parallel to x, the segment NN' of any straight line X, included between x and m, is 

 equal to the radius of gyration with respect to x in the arbitrary direction X ; that 

 is to say, if we take the angle X,r=90— co, we have the radius of gyration, in the 



MM' 

 direction X, =^i— =NN'. We can dispense with the perpendiculars. It is suffi- 

 cient to construct, besides the circle 0, the circle r on 0/as diameter : if a; meets 

 the circle r in the point M, and M/" be drawn cutting the circle in the point M', 

 the segment MM' will be the required radius of gyration. 



2. Let G be a circle passing through 0, and of arbitrary radius f. If through the 

 points A we draw two parallels to BB, and through the points B t-.vo parallels to 

 AA, the diagonals of the rectangle so produced meet (i in two point.? a and n', and 

 the straight lines A A, BB meet the same circle in /3 and /3'. i^et \J be the point 

 of intersection of the chords a a and /3 /3'. 



By means of this jyoint U xoe construct the harycentric axis y, conjugate to any given 

 barycentric axis x. It is only necessary to observe that if x cuts G in the point X, 

 and XU cuts G in the point Y, the straight line OU is the axis y required. This 

 is, in fact, merely the construction for the radius y conjugate to x in the involution 

 of the straight lines 0(A B. aa') ; but these latter are two pairs of conjugate dia- 

 meters of the central ellipse of F, whence &c. 



li. Construction for the central nucleus. Draw any suitable number of straight 

 lines enveloping the contour of F without cutting it. Let I be one of these lines, 

 i. e. a tangent which does not cut elsewhere the contour of F (unless it be convex). 

 Draw through O the axis x parallel to /, and through / the perpendicular to /, 

 which meets /, x, and the circle in the points V, ]M, and INI' respectively. "With 

 centre M and radius INIM' describe a circle, intercepting on x the distance !MK' 



* These antipolars are tangents to the contour of F and parallel to (be conjugate direc- 

 tion of OC ; and we know that the barycentre O is situated on each of the finite s?giuents 

 AA' and BB'. 



t Wo might take O coincident with T ; then /3 coincides with /and /3' with O, and U 

 is the point of intersection of ««' and AA'. 



1870. 3 



